Interval-Valued Intuitionistic Fuzzy Multicriteria Group Decision Making Based on VIKOR and Choquet Integral

Abstract

An effective decision making approach based on VIKOR and Choquet integral is developed to solve multicriteria group decision making problem with conflicting criteria and interdependent subjective preference of decision makers in a fuzzy environment where preferences of decision makers with respect to criteria are represented by interval-valued intuitionistic fuzzy sets. First, an interval-valued intuitionistic fuzzy Choquet integral operator is given. Some of its properties are investigated in detail. The extended VIKOR decision procedure based on the proposed operator is developed for solving the multicriteria group decision making problem where the interactive criteria weight is measured by Shapley value. An illustrative example is given for demonstrating the applicability of the proposed decision procedure for solving the multi-criteria group decision making problem in interval-valued intuitionistic fuzzy environment.

1. Introduction

The increasing complexity of the socioeconomic environments makes it less and less possible for a single decision maker to consider all relevant aspects of a problem. Hence, in order to get a more reasonable decision result, a decision organization, such as the board of directors of a company, which contains a collection of decision makers, is set up explicitly or implicitly to assess the alternatives. The analysis must be extended to account, somehow, for group decision makers, each one potentially exhibiting a unique preference structure, perceiving different consequences, and responding to a diverse array of aspirations [1].

Multiple criteria or attribute decision making (MCDM or MADM) problems is to find the best compromise solution among all feasible alternatives assessed on the basis of multiple criteria or attributes, both quantitative and qualitative. Due to the complex structure of the problem and conflicting nature of the criteria for multi-criteria decision making, trade-offs assessment is one of the most difficult issues in multi-criteria decision making. There may be no solution satisfying all criteria simultaneously. Thus, the solution is a set of noninferior solutions or a compromise solution according to the decision maker’s preferences. By using compromise programming, the compromise solution was established by Yu [2] and Zeleny [3] for a problem with conflicting criteria and it can be helping the decision makers to reach a final solution. The compromise solution is a feasible solution, which is the closest to the ideal, and compromise means an agreement established by mutual concessions. Based on the compromise programming [3], many MCDM ranking methods have been investigated, such as the TOPSIS (technique for order preference by similarity to an ideal solution) method [4], the VIKOR (vlsekriterijumska optimizacija i kompromisno resenje in Serbian, meaning multicriteria optimization and compromise solution) method [5], the PROMETHEE (preference ranking organization method for enrichment evaluations) method [6], and the ELECTRE (elimination et choice translating reality) method [7].

As one of the compromise ranking approaches for multiple criteria decision making problems, The VIKOR method focuses on ranking and selecting from the alternatives with conflicting and different units criteria. The vIKOR method of compromise ranking determines a compromise solution that provides the maximum group utility for the majority and a minimum of individual regret [8] for the opponent. Within the VIKOR method, the compromise ranking could be performed by comparing the measure of closeness to the ideal alternative through the process of ranking and selecting a set of alternatives in the presence of conflicting criteria. The obtained compromise solution could be accepted by the decision makers because it provides a maximum group utility of the majority and a minimum individual regret of the opponent. The compromise solutions could be the base for negotiation, involving the decision makers’ preference by criteria weights and a balance between total and individual satisfaction. The VIKOR has been developed for multi-criteria optimization in complex systems and enjoys a wide acceptance. Opricovic and Tzeng [9] made a comparative analysis of VIKOR and TOPSIS. Furthermore, Opricovic and Tzeng [10] extended the VIKOR method with a stability analysis determining the weight stability intervals and with trade-offs analysis and also compared it deeply with the TOPSIS method, the PROMETHEE method, and the ELECTRE method. According to the comparisons, it is obvious that the VIKOR method has many advantages in handling the MCDM problems especially when they are with conflicting and noncommensurable criteria. Tzeng et al. [11] used and compared the VIKOR and TOPSIS methods in solving a public transportation problem. Sayadi et al. [12] extended the VIKOR method to MADM problem with interval numbers. Chang and Hsu [13] showed that the VIKOR method is advantageous for evaluating the relative environmental vulnerability of subdivisions in a watershed. Up to now, the VIKOR method has been applied widely in many fields, such as mountain destination choosing [9], alternative hydropower systems evaluating [9], forestation and forest preservation [14], postearthquake sustainable reconstruction [15], airlines service quality evaluation [16], and selection of the renewable energy project [17].

In complex decision making problems, imprecision and uncertainty are often involved in decision making process due to incomplete information, abundant information, conflicting evidence, ambiguous information, and subjective information [18–20]. The preference information provided by decision maker is also imprecise or uncertain due to time pressure, lack of data, or the decision maker’s limited attention and information processing capacities. To adequately model such decision making situations, fuzzy set [21] and its higher order extensions such as intuitionistic fuzzy set (IFS) [22] and interval-valued intuitionistic fuzzy set (IVIFS) [23] have been successfully used to handle imprecise and uncertain human decision behaviors [20, 24, 25].

Recently, The VIKOR method has been developed for fuzzy multi-criteria decision making in complex systems. Opricovic and Tzeng [26] suggested using fuzzy logic for the VIKOR method. Büyüközkan and Ruan [27] extended the VIKOR method to effectively solve software evaluation problem under a fuzzy environment. Sanayei et al. [28] proposed a hierarchy MCDM model based on fuzzy sets theory and VIKOR method to deal with the supplier selection problems in the supply chain system. Opricovic [29] proposed fuzzy VIKOR method for water resources planning management. Chen and Wang [30] proposed a more efficient delivery approach for evaluating and assessing possible suppliers/vendors by using the fuzzy VIKOR method. Du and Liu [31] developed three extensions of the VIKOR method based on the expected values of the intuitionistic trapezoidal fuzzy numbers and the distances between the intuitionistic trapezoidal fuzzy numbers and the interval numbers, respectively. Devi [32] extended VIKOR method to an intuitionistic fuzzy environment for robot selection. Park et al. [33] extended the VIKOR method for dynamic intuitionistic fuzzy multiple attribute decision making. Park et al. [34] extended the concept of the VIKOR method to develop a methodology for solving the MCDM problems with interval-valued intuitionistic fuzzy numbers.

It is worthwhile noting that the above different kinds of extensions of the VIKOR are based on the assumption that the criteria and subjective preferences of the decision maker are independent with the use of a linear-based mathematical model for facilitating the human decision making process, which is characterized by an independence axiom [35, 36]. However, for real MCDM problems, there is always some degree of interdependent characteristics among criteria [37–41]. For example, we are to evaluate a set of students in relation to three subjects: {mathematics, physics, literature}; we want to give more importance to science-related subjects than to literature, but on the other hand we want to give some advantage to students that are good both in literature and in any of the science-related subjects [38]. Individual decision makers may come from the same or similar research fields in a given situation. They often have similar knowledge, social status, and preference. As a result, their subjective preferences in the group decision-making process often show some interactive characteristics. Traditional additive aggregation operators such as the weighted average or ordered weighted averaging (OWA) operator [42] are based on an implicit assumption that the criteria or the preferences of individual decision maker is independent of each another. This leads to a wide application of the additive expected utility mode [35] based on the mutual preferential independence [36] for solving the multi-criteria group decision making problem. There are, however, some experiments that show that the additive expected utility model and the assumption of mutual preferential independence are often violated [43], which restricts the potential applications of the traditional additive aggregation operators to more extensive areas.

As the generalization of the weighted average operator and the OWA operator, the Choquet integral with respect to fuzzy measure [41] is an effective tool for extending the additive expected utility model that allow weakening the inherent independence hypotheses for modeling the subjective human decision making process [44]. In order to overcome the drawback of those extensions of the VIKOR, it is necessary to develop a new extension of the VIKOR method for fuzzy group multi-criteria decision making. The aim of this paper is to extend the classical VIKOR method to solve the fuzzy multi-criteria group decision making problems in interval-valued intuitionistic fuzzy environment where all the preference information provided by the decision maker is presented as interval-valued intuitionistic fuzzy set, where inter-dependent or interactive characteristics among criteria and preference of decision makers are also taken into account. The motivation and main features of the proposed extended VIKOR method in this paper are summarized as follows.

(i) Although [34] extended the VIKOR for the multi-criteria group decision making problems in interval-valued intuitionistic fuzzy environment, they did not take the interactive characteristics among preference of decision makers into account. An additive interval-valued intuitionistic fuzzy aggregation operator, the interval-valued intuitionistic fuzzy weighted geometric operator, was used to aggregate decision makers’ preferences. This paper considers the interactive characteristics among decision makers’ preferences, a nonlinear aggregation operator; interval-valued intuitionistic fuzzy Choquet integral operator, is proposed for aggregating decision makers’ preferences where the interactive characteristics among decision-makers’ preferences are expressed by means of fuzzy measure. It is shown that when the fuzzy measure is an additive fuzzy measure, decision-makers’ subjective preferences are independent; the nonlinear aggregation operator is reduced to an additive interval-valued intuitionistic fuzzy aggregation operator.

(ii) When inter-dependent or interactive characteristics among criteria are taken into account, the weights of interactive criteria should not be determined by traditional methods. In this paper, we use fuzzy measure to measure the inter-dependent characteristics among criteria; then Shapley value is used to determine the interactive criteria weights, which reflects decision-maker’s subjective preference to criteria as well as the inherent objective characteristics of criteria. It is shown that when the fuzzy measure is an additive fuzzy measure, the Shapley weights is reduced to the classical weights.

(iii) In the classical VIKOR, the criteria weights and decision-maker’s weights are also given in advance. Furthermore, the classical VIKOR is only appropriate for the decision making problems with the crisp assessment information and cannot solve the fuzzy multi-criteria group decision making with interval-valued intuitionistic fuzzy sets. The fundamental characteristic of the interval-valued intuitionistic fuzzy set is that the value of the membership function and nonmembership function of the interval-valued intuitionistic fuzzy set is interval rather than exact number in [0,1], respectively. The interval-valued intuitionistic fuzzy set is more suitable for dealing with imprecise and uncertain information than fuzzy set or intuitionistic fuzzy set. The extended VIKOR method has more flexible for dealing with fuzzy information.

(iv) Compared with the similar methods, the proposed method in this paper can maximize the group utility and minimize the individual regret, and the inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account simultaneously, which make the decision result more reasonable. The decision-makers can adjust the coefficient of decision mechanism to reflect the different importance of group utility and individual regret, which greatly enhances the flexibility of the proposed method. Moreover, though the proposed method is developed to solve multi-criteria group decision making with interval-valued intuitionistic fuzzy set, it also can be used to solve MCDM with intuitionistic fuzzy set.

The remaining of this paper is organized as follows. We first present some preliminary concepts including fuzzy measure, Choquet integral, and interval-valued intuitionistic fuzzy set in Sections 2 and 3, respectively. An interval-valued intuitionistic fuzzy Choquet integral operator is given, and some of its properties are investigated in detail in Section 4. In Section 5, subjective and objective weights of criteria are proposed according to fuzzy measure and Shapley value. In Section 6, the extended VIKOR decision procedure based on the proposed operator is developed for effectively tackling the multi-criteria group decision making problem in an interval-valued intuitionistic fuzzy environment. In Section 7, we give an example for illustrating the applicability of the proposed decision procedure for solving the real world decision problem. Finally, the paper is concluded with some observations in Section 8.

2. Fuzzy Measure and Choquet Integral

Fuzzy measure (a nonadditive measure) is proposed for addressing the issue of nonadditivity in the traditional additive utility model in decision making. It can be defined as follows [41].

Definition 1. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a universe of discourse and let 𝑃(𝑋) be the power set of 𝑋. A fuzzy measure on 𝑋 is a set function 𝜇∶𝑃(𝑋)→[0,1], satisfying the following conditions:(i)𝜇(∅)=0, 𝜇(𝑋)=1.(ii)If 𝐴,𝐵∈𝑃(𝑋) and 𝐴⊆𝐵, then 𝜇(𝐴)≤𝜇(𝐵).
In the multi-criteria decision making, 𝜇(𝐴) can be viewed as the grade of subjective importance of decision criteria set 𝐴. Thus, in addition to the usual weights on criteria taken separately, weights on any combination of criteria are also defined. To determine such fuzzy measures, it needs to find 2^𝑛−2 values for 𝑛 criteria in a given multi-criteria group decision making situation. With the definition as above, the values of 𝜇(∅) and 𝜇(𝑋) are always equal to 0 and 1, respectively. It is obvious that such an evaluation process is quite complex with the structure being difficult to grasp. To reduce the computational complexity, the 𝜆-fuzzy measure 𝑔 that acts as a special kind of fuzzy measure is proposed which is defined as follows [45].

Definition 2. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a universe of discourse; a fuzzy measure 𝑔 on 𝑋 is called 𝜆-fuzzy measure if it satisfies the following conditions: 𝑔(𝐴∪𝐵)=𝑔(𝐴)+𝑔(𝐵)+𝜆𝑔(𝐴)𝑔(𝐵), (1) where 𝜆∈(−1,∞) for all 𝐴, 𝐵∈𝑃(𝑋) and 𝐴∩𝐵=∅.

When 𝜆=0, 𝑔(𝐴∪𝐵)=𝑔(𝐴)+𝑔(𝐵). This means that 𝐴 and 𝐵 are mutually independent, that is; 𝑔 is additive measure and there exists no interaction between 𝐴 and 𝐵. 𝜆≠0 indicates that 𝑔 is nonadditive and there exist some degrees of interdependency between 𝐴 and 𝐵. If 𝜆>0, 𝑔(𝐴∪𝐵)>𝑔(𝐴)+𝑔(𝐵). This implies that the set {𝐴,𝐵} has multiplicative effect. If 𝜆<0,  𝑔(𝐴∪𝐵)<𝑔(𝐴)+𝑔(𝐵). This shows that the set {𝐴,𝐵} has substitutive effect. With the use of parameter 𝜆, the interaction between criteria in multi-criteria decision making can be adequately represented.

If 𝑋 is a finite set, ⋃^𝑛_𝑖=1𝑥_𝑖=𝑋. The 𝜆-fuzzy measure 𝑔 satisfies the following equation: 𝑔(𝑋)=𝑔^𝑛_𝑖=1𝑥_𝑖=1𝜆^𝑛_𝑖=11+𝜆𝑔𝑥_𝑖−1,if𝜆≠0,_𝑛𝑖=1𝑔𝑥_𝑖,if𝜆=0, (2) where 𝑥_𝑖∩𝑥_𝑗=∅ for all 𝑖,𝑗=1,2,…,𝑛 and 𝑖≠𝑗. 𝑔(𝑥_𝑖) for a subset with a single element 𝑥_𝑖 is called a fuzzy density, denoted by 𝑔_𝑖=𝑔(𝑥_𝑖).

Especially for every subset 𝐴∈𝑃(𝑋), we have 𝑔(𝐴)=1𝜆_𝑖∈𝐴1+𝜆𝑔(𝑖)−1,if𝜆≠0,_𝑖∈𝐴𝑔(𝑖),if𝜆=0. (3) The value 𝜆 can be uniquely determined based on (2) from 𝑔(𝑋)=1 by solving the equation 𝜆+1=^𝑛_𝑖=11+𝜆𝑔_𝑖. (4) It should be noted that 𝜆 can also be uniquely determined by 𝑔(𝑋)=1.

As a generalization of the linear Lebesgue integral, the Choquet integral is defined as follows [41].

Definition 3. Let 𝑓 be a positive real-valued function on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. The discrete Choquet integral of 𝑓 with respect to 𝜇 is defined by 𝐶_𝜇𝑓=^𝑛_𝑖=1𝑓𝑥_(𝑖)𝜇𝐴_(𝑖)−𝜇𝐴_(𝑖+1), (5) where the subscript (⋅) indicates a permutation on 𝑋 such that 𝑓(𝑥₍₁₎)≤𝑓(𝑥₍₂₎)≤⋯≤𝑓(𝑥_(𝑛)). Also, 𝐴_(𝑖)={𝑥_(𝑖),…,𝑥_(𝑛)}, and 𝐴_(𝑛+1)=∅.

The Choquet integral is popular in tackling real decision making problems as it coincides with the Lebesgue integral in which the measure is additive. An additive measure may be directly tied to the notions of the additive expected utility theory and mutual preferential independence [36]. The Choquet integral is able to aggregate the criteria weighting information even when the mutual preferential independence is violated [44].

3. Interval-Valued Intuitionistic Fuzzy Set

Let 𝑋 be a universe of discourse; a fuzzy set 𝐴 in 𝑋 can be expressed as 𝐴=𝑥,𝑡_𝐴(𝑥)∣𝑥∈𝑋, (6) where 𝑡_𝐴∶𝑋→[0,1] is a membership function which characterizes the degree of membership of the element 𝑥 in the set 𝐴. The main characteristic of fuzzy sets is that a membership function value is assigned to each element 𝑥 in a universe of discourse 𝑋. A nonmembership degree equals to one minus the membership degree [21]. This means that this single membership degree combines the evidence for 𝑥and the evidence against 𝑥 without indicating how much there is for each. This single membership value therefore tells us nothing about the lack of knowledge in describing the element 𝑥. In real applications, however, the information about an object corresponding to a fuzzy concept may be incomplete. That means that the sum of the membership degree and the nonmembership degree of that object in a universe of discourse corresponding to a fuzzy concept may be less than one. In the fuzzy sets theory, there is no means to incorporate the lack of knowledge with the membership degree. To effectively deal with this issue, intuitionistic fuzzy set [22] is introduced as an extension of the traditional fuzzy set as follows.

Definition 4. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a universe of discourse; an intuitionistic fuzzy set in 𝑋 is an expression 𝐴 given by 𝐴=𝑥,𝑡_𝐴(𝑥),𝑓_𝐴(𝑥)∣𝑥∈𝑋, (7) where 𝑡_𝐴∶𝑋→[0,1], 𝑓_𝐴∶𝑋→[0,1] with the condition: 0≤𝑡_𝐴(𝑥)+𝑓_𝐴(𝑥)≤1, for all 𝑥 in 𝑋. The numbers 𝑡_𝐴(𝑥) and 𝑓_𝐴(𝑥) represent the degree of membership and the degree of nonmembership of the element 𝑥 in the set 𝐴, respectively.

For each intuitionistic fuzzy set 𝐴 in 𝑋, if 𝜋_𝐴(𝑥)=1−𝑡_𝐴(𝑥)−𝑓_𝐴(𝑥), for all 𝑥∈𝑋, then 𝜋_𝐴(𝑥) is called the degree of indeterminacy of 𝑥 to 𝐴. Especially, if 𝜋_𝐴(𝑥)=1−𝑡_𝐴(𝑥)−𝑓_𝐴(𝑥)=0, for all 𝑥∈𝑋, the intuitionistic fuzzy set 𝐴 is reduced to a fuzzy set.

Interval-valued intuitionistic fuzzy set [23] is a generalization of the intuitionistic fuzzy sets. The fundamental characteristic of the interval-valued intuitionistic fuzzy set is that the value of the membership function and non-membership function of the interval-valued intuitionistic fuzzy set is interval rather than exact number.

Definition 5. Let 𝑋 be a universe of discourse and let 𝐷[0,1] be the set of all closed subintervals of the interval [0,1]. An interval-valued intuitionistic fuzzy set 𝐴 in 𝑋 is an expression given by 𝐴=𝑥,𝑡_𝐴(𝑥),𝑓_𝐴(𝑥)∣𝑥∈𝑋, (8) where 𝑡_𝐴∶𝑋→𝐷[0,1], 𝑓_𝐴∶𝑋→𝐷[0,1] with the condition 0≤sup𝑡_𝐴(𝑥)+sup𝑓_𝐴(𝑥)≤1. The intervals 𝑡_𝐴(𝑥) and 𝑓_𝐴(𝑥) denote, respectively, the degree of belongingness and the degree of nonbelongingness of the element 𝑥 to the set 𝐴.

For any two intervals [𝑎,𝑏] and [𝑐,𝑑] with 𝑏+𝑑≤1 belonging to 𝐷[0,1], let 𝑡_𝐴(𝑥)=[𝑎,𝑏], and 𝑓_𝐴(𝑥)=[𝑐,𝑑]. An interval-valued intuitionistic fuzzy set can be denoted by 𝐴={⟨𝑥,[𝑎,𝑏],[𝑐,𝑑]⟩∣𝑥∈𝑋}. In this paper, ([𝑎,𝑏],[𝑐,𝑑]) is referred as an interval-valued intuitionistic fuzzy value. For convenience, let Ω be the set of all intervalvalued intuitionistic fuzzy values on 𝑋. Obviously, 𝑎⁺=([1,1],[0,0]) and 𝑎⁻=([0,0],[1,1]) are the largest and smallest interval-valued intuitionistic fuzzy values, respectively. A distance measure between inter-valued intuitionistic fuzzy values is defined as follows [24].

Definition 6. Let 𝑋={𝑥₁,…,𝑥_𝑛} be a universe of discourse, and 𝑎=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) and 𝑏̃=([𝑎^′_𝑖,𝑏^′_𝑖],[𝑐^′_𝑖,𝑑^′_𝑖])(𝑖=1,…,𝑛) be two interval-valued intuitionistic fuzzy values on 𝑋; 𝑑𝑏̃=1𝑎,4^𝑛_𝑖=1𝑎_𝑖−𝑎^′_𝑖+𝑏_𝑖−𝑏^′_𝑖+𝑐_𝑖−𝑐^′_𝑖+𝑑_𝑖−𝑑^′_𝑖 (9) is called the normalized Hamming distance between 𝑎 and 𝑏̃, if 𝑑𝑏̃=1𝑎,4^𝑛_𝑖=1𝑤_𝑖𝑎_𝑖−𝑎^′_𝑖+𝑏_𝑖−𝑏^′_𝑖+𝑐_𝑖−𝑐^′_𝑖+𝑑_𝑖−𝑑^′_𝑖, (10) where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weight vector of 𝑥_𝑗 such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, and 𝑑(𝑎,𝑏̃) is called the weighted Hamming distance between 𝑎 and 𝑏̃.

The following expressions are defined for any two interval-valued intuitionistic fuzzy values 𝑎₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝑎₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) [24]:(i)𝑎₁≤𝑎₂iff𝑏₁≤𝑏₂,𝑎₁≤𝑎₂,𝑑₁≥𝑑₂,𝑐₁≥𝑐₂, (11)(ii)𝑎₁=𝑎₂iff𝑏₁=𝑏₂,𝑎₁=𝑎₂,𝑑₁=𝑑₂,𝑐₁=𝑐₂. (12)

Equation (11), however, is not always satisfied in some decision making situations. To address this issue, the following order relation between interval-valued intuitionistic fuzzy values is defined [46].

Definition 7. Let 𝑎₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝑎₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be two interval-valued intuitionistic fuzzy values, let 𝑆(𝑎₁)=(𝑎₁−𝑐₁+𝑏₁−𝑑₁)/2 and 𝑆(𝑎₂=(𝑎₂−𝑐₂+𝑏₂−𝑑₂)/2 be the score functions of 𝑎₁ and 𝑎₂, respectively, and let 𝐻(𝑎₁)=(𝑎₁+𝑐₁+𝑏₁+𝑑₁)/2 and 𝐻(𝑎₂)=(𝑎₂+𝑐₂+𝑏₂+𝑑₂)/2 be the accuracy functions of 𝑎₁ and 𝑎₂, respectively:(a)if 𝑆(𝑎₁)<𝑆(𝑎₂), 𝑎₁ is smaller than 𝑎₂, denoted by 𝑎₁<𝑎₂;(b)if 𝑆(𝑎₁)=𝑆(𝑎₂):(i)if 𝐻(𝑎₁)<𝐻(𝑎₂), 𝑎₁ is smaller than 𝑎₂, denoted by 𝑎₁<𝑎₂;(ii)if 𝐻(𝑎₁)=𝐻(𝑎₂), 𝑎₁ and 𝑎₂ represent the same information, denoted by 𝑎₁=𝑎₂.

4. Interval-Valued Intuitionistic Fuzzy Choquet Integral Operator

In interval-valued intuitionistic fuzzy multi-criteria group decision making problems, some additive linear operators, such as interval-valued intuitionistic fuzzy weighted averaging [46], interval-valued intuitionistic fuzzy weighted geometric [47, 48], or interval-valued intuitionistic fuzzy OWA [48] operators, are applied for aggregating the decision maker’s preference information. Existing research for multi-criteria group decision making does not adequately address the issue of interdependencies among criteria or decision-makers’ preference. Inspired by the idea of the Choquet integral operator in decision making, in this section, an interval-valued intuitionistic fuzzy Choquet integral operator is given for effectively addressing the issue of interdependence among decision-makers’ preference in a fuzzy environment [49, 50]. And some of its properties are investigated in detail. The relation between this operator and the additive linear interval-valued intuitionistic fuzzy aggregation operators is analyzed. To do this, first two operational laws for interval-valued intuitionistic fuzzy values [24, 46] are introduced.

Definition 8. Let 𝑎₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝑎₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be two interval-valued intuitionistic fuzzy values;(i)𝑎₁+𝑎₂=([𝑎₁+𝑎₂−𝑎₁𝑎₂,𝑏₁+𝑏₂−𝑏₁𝑏₂],[𝑐₁𝑐₂,𝑑₁𝑑₂]);(ii)𝜆𝑎₁=([1−(1−𝑎₁)^𝜆,1−(1−𝑏₁)^𝜆],[𝑐^𝜆₁,𝑑^𝜆₁]), 𝜆>0.
These two operational laws have two properties described as follows.

Proposition 9. Let 𝑎₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝑎₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be two interval-valued intuitionistic fuzzy values, and let 𝑐̃=𝑎₁+𝑎₂ and 𝑑=𝜆𝑎₁; both 𝑐̃ and 𝑑 are also interval-valued intuitionistic fuzzy values.

Proposition 10. Let 𝑎₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]) and 𝑎₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be two interval-valued intuitionistic fuzzy values, for all 𝜆₁, 𝜆₂>0, (i)𝑎₁+𝑎₂=𝑎₂+𝑎₁;(ii)𝜆₁(𝑎₁+𝑎₂)=𝜆₁𝑎₁+𝜆₁𝑎₂;(iii)𝜆₁𝑎₁+𝜆₂𝑎₁=(𝜆₁+𝜆₂)𝑎₁.

According to these operational laws, an interval-valued intuitionistic fuzzy Choquet integral operator is defined as follows [49, 51].

Definition 11. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. The discrete interval-valued intuitionistic fuzzy Choquet integral (I-IFC) of 𝑎_𝑖 with respect to 𝜇 is defined by I-IFC_𝜇𝑎₁,…,𝑎_𝑛=𝑎₍₁₎𝜇𝐴₍₁₎−𝜇𝐴₍₂₎+𝑎₍₂₎𝜇𝐴₍₂₎−𝜇𝐴₍₃₎+⋯+𝑎_(𝑛)𝜇𝐴_(𝑛)−𝜇𝐴_(𝑛+1)=^𝑛_𝑖=1𝑎_(𝑖)𝜇𝐴_(𝑖)−𝜇𝐴_(𝑖+1). (13) Further, the aggregated value is also an interval-valued intuitionistic fuzzy value, and I-IFC_𝜇𝑎₁,…,𝑎_𝑛=1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, (14) where the subscript (⋅) indicates a permutation on X such that 𝑎₍₁₎≤𝑎₍₂₎≤⋯≤𝑎_(𝑛). And 𝐴_(𝑖)=((𝑖),…,(𝑛)), 𝐴_(𝑛+1)=∅.

Remark 12. Obviously if interval-valued intuitionistic fuzzy value 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) is reduced to intuitionistic fuzzy values, the interval-valued intuitionistic fuzzy Choquet integral of 𝑎_𝑖 with respective to 𝜇 is reduced to an intuitionistic fuzzy Choquet integral operator [52].

Remark 13. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) and 𝑏̃_𝑖=([𝑎^′_𝑖,𝑏^′_𝑖],[𝑐^′_𝑖,𝑑^′_𝑖])(𝑖=1,2,…,𝑛) be two collections of interval-valued intuitionistic fuzzy values on 𝑋. Since 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑎^′_𝑖,𝑏^′_𝑖,𝑐^′_𝑖,𝑑^′_𝑖∈[0,1] for any 𝑖, if we assume that 𝑇_𝑃(𝑐_𝑖,𝑐^′_𝑖)=𝑐_𝑖𝑐^′_𝑖, 𝑇_𝑃(𝑑_𝑖,𝑑^′_𝑖)=𝑑_𝑖𝑑^′_𝑖, 𝑆_𝑃(𝑎_𝑖,𝑎^′_𝑖)=𝑎_𝑖+𝑎^′_𝑖−𝑎_𝑖𝑎^′_𝑖, and 𝑆_𝑃(𝑏_𝑖,𝑏^′_𝑖)=𝑏_𝑖+𝑏^′_𝑖−𝑏_𝑖𝑏^′_𝑖, then 𝑇_𝑃(𝑐_𝑖,𝑐^′_𝑖) and 𝑇_𝑃(𝑑_𝑖,𝑑^′_𝑖)  are two of the basic 𝑡-norms, called the product, which is satisfying the following properties [53]: 𝑇_𝑃(𝑥,1)=𝑥 (boundary); 𝑇_𝑃(𝑥,𝑦)≤𝑇_𝑃(𝑥,𝑧) whenever 𝑦≤𝑧 (monotonicity); 𝑇_𝑃(𝑥,𝑦)=𝑇_𝑃(𝑦,𝑥) (commutativity); and 𝑇_𝑃(𝑥,𝑇_𝑃(𝑦,𝑧))=𝑇_𝑃(𝑇_𝑃(𝑥,𝑦)) (associativity), where 𝑥,𝑦,𝑧∈[0,1]. 𝑆_𝑃(𝑎_𝑖,𝑎^′_𝑖) and 𝑆_𝑃(𝑏_𝑖,𝑏^′_𝑖)  are two of the basic 𝑡-conorms, called the probabilistic sum, and 𝑆_𝑃 is also called the dual 𝑡-conorm of 𝑇_𝑃, which is satisfying the boundary; that is, 𝑆_𝑃(𝑥,0)=𝑥, monotonicity, commutativity, and associativity [53]. The associativity of 𝑡-norms and 𝑡-conorms allows us to extend the product 𝑇_𝑃 and probabilistic sum 𝑆_𝑃 in unique way to an 𝑛-ary operation in the usual way by induction, defining for each 𝑛-tuple (𝑥₁,𝑥₂,…,𝑥_𝑛)∈[0,1]^𝑛 and (𝑦₁,𝑦₂,…,𝑦_𝑛)∈[0,1]^𝑛, respectively, 𝑇_𝑃𝑥₁,𝑥₂,…,𝑥_𝑛=𝑇^𝑛_𝑃𝑖=1𝑥_𝑖=𝑇𝑇_𝑃^𝑛−1_𝑃𝑖=1𝑥_𝑖,𝑥_𝑛=^𝑛_𝑖=1𝑥_𝑖,𝑆_𝑃𝑦₁,𝑦₂,…,𝑦_𝑛=𝑆^𝑛_𝑃𝑖=1𝑦_𝑖=𝑆𝑆_𝑃^𝑛−1_𝑃𝑖=1𝑦_𝑖,𝑦_𝑛=1−^𝑛_𝑖=11−𝑦_𝑖. (15) Assume that 𝑦_𝑖=1−(1−𝑎_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, 𝑦^′_𝑖=1−(1−𝑏_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, and 𝑥_𝑖=(𝑐_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, 𝑥^′_𝑖=(𝑑_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, I-IFC_𝜇𝑎₁,…,𝑎_𝑛=𝑆_𝑃𝑦₁,…,𝑦_𝑛,𝑆_𝑃𝑦^′₁,…,𝑦^′_𝑛,𝑇_𝑃𝑥₁,…,𝑥_𝑛,𝑇_𝑃𝑥^′₁,…,𝑥^′_𝑛. (16) This means that the interval-valued intuitionistic fuzzy Choquet integral operator can be represented by one of the basic 𝑡-norms 𝑇_𝑃 and 𝑡-conorms 𝑆_𝑃.

In the following, we discuss some desirable properties about the interval-valued intuitionistic fuzzy Choquet integral operator.

Proposition 14. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. If all 𝑎_𝑖(𝑖=1,2,…,𝑛) are equal; that is, for all 𝑖, 𝑎_𝑖=𝑎=([𝑎,𝑏],[𝑐,𝑑]), 𝐼-𝐼𝐹𝐶_𝜇(𝑎₁,…,𝑎_𝑛)=𝑎.

Proof. According to (14), we have if for all 𝑖(𝑖=1,2,…,𝑛), 𝑎_𝑖=𝑎, I-IFC_𝜇𝑎₁,…,𝑎_𝑛=1−(1−𝑎)^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−(1−𝑏)^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑐^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑑^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}. (17) Since ∑^𝑛_𝑖=1(𝜇(𝐴_(𝑖))−𝜇(𝐴_(𝑖+1)))=1, we have that I-IFC_𝜇𝑎₁,…,𝑎_𝑛=([𝑎,𝑏],[𝑐,𝑑])=𝑎. (18)

Proposition 15. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) and 𝑏̃_𝑖=([𝑎^′_𝑖,𝑏^′_𝑖],[𝑐^′_𝑖,𝑑^′_𝑖])(𝑖=1,2,…,𝑛) be two collections of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. The subscript (⋅) indicates a permutation such that 𝑎₍₁₎≤⋯≤𝑎_(𝑛) and 𝑏̃₍₁₎≤⋯≤𝑏̃_(𝑛). If 𝑏_(𝑖)≤𝑏^′_(𝑖), 𝑎_(𝑖)≤𝑎^′_(𝑖) and 𝑑_(𝑖)≥𝑑^′_(𝑖), 𝑐_(𝑖)≥𝑐^′_(𝑖) for all 𝐼; that is, 𝑎_(𝑖)≤𝑏̃_(𝑖), 𝐼-𝐼𝐹𝐶_𝜇𝑎₁,…,𝑎_𝑛≤𝐼-𝐼𝐹𝐶_𝜇𝑏̃₁,…,𝑏̃_𝑛. (19)

Proof. Since 𝐴_(𝑖+1)⊆𝐴_(𝑖), then 𝜇(𝐴_(𝑖))−𝜇(𝐴_(𝑖+1))≥0. For all 𝑖,  𝑏_(𝑖)≤𝑏^′_(𝑖), 𝑎_(𝑖)≤𝑎^′_(𝑖) and 𝑑_(𝑖)≥𝑑^′_(𝑖), 𝑐_(𝑖)≥𝑐^′_(𝑖), 1−∏^𝑛_𝑖=1(1−𝑎_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−∏^𝑛_𝑖=1(1−𝑎^′_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, 1−∏^𝑛_𝑖=1(1−𝑏_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−∏^𝑛_𝑖=1(1−𝑏^′_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, ∏^𝑛_𝑖=1(𝑓𝑎_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≥∏^𝑛_𝑖=1̃𝑏(𝑓_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}, ∏^𝑛_𝑖=1(𝑑_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≥∏^𝑛_𝑖=1(𝑑^′_(𝑖))^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}.
According to (14) and (11), we have I-IFC_𝜇𝑎₁,…,𝑎_𝑛≤I-IFC_𝜇𝑏̃₁,…,𝑏̃_𝑛. (20)

Proposition 16. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. If 𝑎⁻=min_𝑖𝑎_𝑖,min_𝑖𝑏_𝑖,max_𝑖𝑐_𝑖,max_𝑖𝑑_𝑖,𝑎⁺=max_𝑖𝑎_𝑖,max_𝑖𝑏_𝑖,min_𝑖𝑐_𝑖,min_𝑖𝑑_𝑖, (21) then 𝑎⁻≤𝐼-𝐼𝐹𝐶_𝜇(𝑎₁,…,𝑎_𝑛)≤𝑎⁺.

Proof. For any 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,…,𝑛), it is obvious that 𝑎⁻and 𝑎⁺are interval-valued intuitionistic fuzzy values. Since 𝐴_(𝑖+1)⊆𝐴_(𝑖), then 𝜇(𝐴_(𝑖))−𝜇(𝐴_(𝑖+1))≥0. Let the subscript (⋅) indicate a permutation such that 𝑎₍₁₎≤⋯≤𝑎_(𝑛); for all 𝑖, we have min_𝑖𝑎_𝑖≤𝑎_(𝑖)≤max_𝑖𝑎_𝑖,min_𝑖𝑏_𝑖≤𝑏_(𝑖)≤max_𝑖𝑏_𝑖,min_𝑖𝑐_𝑖≤𝑐_(𝑖)≤max_𝑖𝑐_𝑖,min_𝑖𝑑_𝑖≤𝑑_(𝑖)≤max_𝑖𝑑_𝑖. (22) Thus, 1−^𝑛_𝑖=11−min_𝑖𝑎_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−max_𝑖𝑎_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−^𝑛_𝑖=11−min_𝑖𝑏_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−max_𝑖𝑏_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1min_𝑖𝑐_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1max_𝑖𝑐_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1min_𝑖𝑑_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1max_𝑖𝑑_𝑖^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}; (23) that  is, 1−1−min_𝑖𝑎_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−1−max_𝑖𝑎_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−1−min_𝑖𝑏_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤1−1−max_𝑖𝑏_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},min_𝑖𝑐_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤max_𝑖𝑐_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},min_𝑖𝑑_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤^𝑛_𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤max_𝑖𝑑_𝑖^{∑𝑛𝑖=1𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}. (24) So we have min_𝑖𝑎_𝑖≤1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤max_𝑖𝑎_𝑖,min_𝑖𝑏_𝑖≤1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤max_𝑖𝑏_𝑖,min_𝑖𝑐_𝑖≤^𝑛_𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤max_𝑖𝑐_𝑖,min_𝑖𝑑_𝑖≤^𝑛_𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}≤min_𝑖𝑑_𝑖. (25) According to (14) and (11), we have min_𝑖𝑎_𝑖,min_𝑖𝑏_𝑖,max_𝑖𝑐_𝑖,max_𝑖𝑑_𝑖≤I-IFC_𝜇𝑎₁,…,𝑎_𝑛≤max_𝑖𝑎_𝑖,max_𝑖𝑏_𝑖,min_𝑖𝑐_𝑖,min_𝑖𝑑_𝑖; (26) that is, 𝑎⁻≤I-IFC_𝜇(𝑎₁,…,𝑎_𝑛)≤𝑎⁺.

Proposition 17. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. If 𝑠̃=([𝑎,𝑏],[𝑐,𝑑]) is an interval-valued intuitionistic fuzzy value on 𝑋, then 𝐼-𝐼𝐹𝐶_𝜇𝑎₁+𝑠,̃…,𝑎_𝑛+𝑠̃=𝐼-𝐼𝐹𝐶_𝜇𝑎₁,…,𝑎_𝑛+𝑠.̃ (27)

Proof. Since for any 𝑖(𝑖=1,2,…,𝑛), 𝑎_𝑖+𝑠̃=𝑎_𝑖+𝑎−𝑎_𝑖𝑎,𝑏_𝑖+𝑏−𝑏_𝑖𝑏,𝑐_𝑖𝑐,𝑑_𝑖𝑑=1−1−𝑎_𝑖(1−𝑎),1−1−𝑏_𝑖(1−𝑏),𝑐_𝑖𝑐,𝑑_𝑖𝑑. (28) According to (14), we have I-IFC_𝜇𝑎₁+𝑠,̃…,𝑎_𝑛+𝑠̃=1−^𝑛_𝑖=11−𝑎_(𝑖)(1−𝑎)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−^𝑛_𝑖=11−𝑏_(𝑖)(1−𝑏)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑐_(𝑖)𝑐^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑑_(𝑖)𝑑^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}=1−(1−𝑎)^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−(1−𝑏)^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑐^𝑛_𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑑^𝑛_𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}. (29) According to Definition 8, we have I-IFC_𝜇𝑎₁,…,𝑎_𝑛+𝑠̃=1−(1−𝑎)^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−(1−𝑏)^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑐^𝑛_𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},𝑑^𝑛_𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}. (30) Thus, I-IFC_𝜇𝑎₁+𝑠,̃…,𝑎_𝑛+𝑠̃=I-IFC_𝜇𝑎₁,…,𝑎_𝑛+𝑠.̃ (31)

Proposition 18. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])  (𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. If 𝑟>0; then 𝐼-𝐼𝐹𝐶_𝜇𝑟𝑎₁,…,𝑟𝑎_𝑛=𝑟𝐼-𝐼𝐹𝐶_𝜇𝑎₁,…,𝑎_𝑛. (32)

Proof. According to Definition 8, for any 𝑖(𝑖=1,2,…,𝑛) and 𝑟>0, we have 𝑟𝑎_𝑖=1−1−𝑎_𝑖^𝑟,1−1−𝑏_𝑖^𝑟,𝑐_𝑖^𝑟,𝑑_𝑖^𝑟. (33) According to (14), we have I-IFC_𝜇𝑟𝑎₁,…,𝑟𝑎_𝑛=1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝑟𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝑟𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑐_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},_𝑛𝑖=1𝑑_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))}=1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},_𝑛𝑖=1𝑐_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},_𝑛𝑖=1𝑑_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))}. (34) Since 𝑟I-IFC_𝜇𝑎₁,…,𝑎_𝑛=𝑟1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))},_𝑛𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))}=1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))𝑟},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))𝑟},_𝑛𝑖=1𝑐_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))𝑟},_𝑛𝑖=1𝑑_(𝑖)^{𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1))𝑟}=1−^𝑛_𝑖=11−𝑎_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},1−^𝑛_𝑖=11−𝑏_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},_𝑛𝑖=1𝑐_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))},_𝑛𝑖=1𝑑_(𝑖)^{𝑟(𝜇(𝐴(𝑖))−𝜇(𝐴(𝑖+1)))}, (35) thus, I-IFC_𝜇(𝑟𝑎₁,…,𝑟𝑎_𝑛)=𝑟I-IFC_𝜇(𝑎₁,…,𝑎_𝑛).

According to Propositions 17 and 18, the following corollary can be derived.

Corollary 19. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. If 𝑟>0 and 𝑠̃=([𝑎,𝑏],[𝑐,𝑑]) is an interval-valued intuitionistic fuzzy value on 𝑋, 𝐼-𝐼𝐹𝐶_𝜇𝑟𝑎₁+𝑠,̃…,𝑟𝑎_𝑛+𝑠̃=𝑟𝐼-𝐼𝐹𝐶_𝜇𝑎₁,…,𝑎_𝑛+𝑠.̃ (36)

Remark 20. From the above analysis, the interval-valued intuitionistic fuzzy Choquet integral operator has the same properties as these of the intuitionistic fuzzy Choquet integral operator [52].

Definition 21. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋. An interval-valued intuitionistic fuzzy ordered weighted averaging (I-IFOWA) operator of dimension 𝑛 is a mapping I-IFOWA: Ω^𝑛→Ω, that has an associated weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, and I-IFOWA_𝑤𝑎₁,𝑎₂,…,𝑎_𝑛=𝑤₁𝑎₍₁₎+𝑤₂𝑎₍₂₎+⋯+𝑤_𝑛𝑎_(𝑛), (37) where the subscript (⋅) indicates a permutation such that 𝑎₍₁₎≤⋯≤𝑎_(𝑛). Furthermore, I-IFOWA_𝑤𝑎₁,…,𝑎_𝑛=1−^𝑛_𝑖=11−𝑎_(𝑖)^𝑤𝑖,1−^𝑛_𝑖=11−𝑏_(𝑖)^𝑤𝑖,_𝑛𝑖=1𝑐_(𝑖)^𝑤𝑖,^𝑛_𝑖=1𝑑_(𝑖)^𝑤𝑖. (38)

Theorem 22. Let 𝑎_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of interval-valued intuitionistic fuzzy values on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. Any interval-valued intuitionistic fuzzy ordered weighted averaging operator with the weighted vector 𝑤=(𝑤₁,…,𝑤_𝑛) is a special interval-valued intuitionistic fuzzy Choquet integral operator, whose fuzzy measure 𝜇 is defined by 𝜇(𝑆)=∑^𝑛_{𝑖=𝑛−𝑠+1}𝑤_𝑖(𝑆⊆𝑋,𝑆≠∅) where 𝑛, 𝑠 is denoted the cardinality of 𝑆, 𝑋, respectively. Reciprocally, any commutative interval-valued intuitionistic fuzzy Choquet integral operator is such that 𝜇(𝑆) depends only on 𝑠, where 𝑠 denotes the cardinality of 𝑆  (𝑆⊆𝑋) and coincides with an interval-valued intuitionistic fuzzy ordered weighted averaging operator, whose weights are 𝑤_𝑛−𝑠=𝜇(𝑆∪𝑖)−𝜇(𝑆), 𝑖∈𝑋, 𝑆⊆𝑋⧵𝑖.

The proof of Theorem 22 is similar to that of Proposition 12 in [52].

Remark 23. The analysis above shows that the relationship between the I-IFC operator and the I-IFOWA operator is similar to that between Choquet integral and OWA operator. Furthermore, If 𝜇 is an additive fuzzy measure on 𝑋 such that 𝜇(𝐴)=∑_𝑖∈𝐴𝑤_𝑖(𝐴⊆𝑋), the interval-valued intuitionistic fuzzy Choquet operator is reduced to the interval-valued intuitionistic fuzzy weighted averaging operator [46]. This means that the interval-valued intuitionistic fuzzy Choquet integral operator generalizes both the interval-valued intuitionistic fuzzy ordered weighted averaging and the interval-valued intuitionistic fuzzy weighted averaging operators. There are interactive or interdependent characteristics among decision making criteria or subjective preferences of individual decision makers; the proposed operator not only can overcome the limitation of these traditional interval-valued intuitionistic fuzzy aggregation operators. But also can be used for better modeling the uncertain and imprecise information in the decision making process.

5. Shapley Weights

In multi-criteria decision making problems, weighting coefficients (weight 𝑤_𝑖) are introduced to express the relative importance of different criteria. Each criterion is endowed with a specific importance weight. Generally, it is seen that weighting methods are categorized into two categories: subjective methods and objective methods [54]. Subjective methods determine weights solely based on the preference or judgments of decision makers. However, objective methods utilize mathematical models, automatically without considering the decision makers’ preferences.

The VIKOR method was developed as a multi-criteria decision making method to solve a discrete decision problem with conflicting criteria [9]. For conflicting criteria, the overall importance of a criterion 𝑖∈𝑁 (𝑁 denotes a criteria set) is not solely determined by itself 𝑖 but also by all other criteria 𝑇, 𝑖∉𝑇. Suppose that 𝑤(𝑖) denotes the importance degree of 𝑖. 𝑤(𝑖)=0 suggests that the element is unimportant. For conflicting criteria, however, it may happen that for many subsets 𝑇⊆𝑁, 𝑤(𝑇∪𝑖) is much greater than 𝑤(𝑇), suggesting that 𝑖 is actually an important criterion in the decision making process. So for decision making problems with conflicting criteria, we consider not only subjective weight of criteria but also objective weight. The fuzzy measure defined as above is capable of modeling this situation, which represents a weight on not only each criterion but also each combination of criteria so that the total of all the 𝑤_𝑖(𝑖=1,2,…,𝑛) does not necessarily equal to one. As a result, the conflicting phenomena between criteria can be represented.

Taking this case into consideration in 1953, Shapley [55] proposed a definition of a coefficient of importance, based on a set of reasonable axioms. The importance index or Shapley value of criterion 𝑖 with respect to fuzzy measure 𝜇 is defined by 𝜙_𝑖𝜇=_{𝑇⊆𝑁⧵𝑖}(𝑛−𝑡−1)!𝑡!𝜇()(𝑇,𝑛!𝑇∪𝑖−𝜇) (39) where 𝑛 and 𝑡 denote the cardinality of set 𝑁, and 𝑇, respectively. Further, according to (3), for 𝜆-fuzzy measure 𝑔, the importance index or Shapley value 𝜙_𝑖(𝑔) of criterion 𝑖 can be expressed by 𝜙_𝑖𝑔=_{𝑇⊆𝑁⧵𝑖}(𝑛−𝑡−1)!𝑡!𝑔(𝑖)𝑛!_𝑗∈𝑇1+𝜆𝑔𝑗, (40) where fuzzy density, 𝑔(𝑖), is a subjective assessment or preference of decision maker to criterion 𝑖, which can be seen as a subjective weight of criterion 𝑖.

A basic property of the Shapley value is ^𝑛_𝑖=1𝜙_𝑖𝜇=1, (41) which mean that {𝜙_𝑖(𝜇)}_𝑖∈𝑁 is a weight vector. When 𝜇 is additive, we clearly have 𝜙_𝑖𝜇=𝜇(𝑖)=𝑤_𝑖,𝑖∈𝑁. (42) In this case decision maker’s subjective preference is only taken into account in the weighting coefficients.

If 𝜇 is nonadditive then some criteria are dependent and (42) generally does not hold anymore. This shows that it is sensible to search for a coefficient of overall importance for each criterion, where the importance index or Shapley value takes into account both decision makers’ subjective preference and the objective weighting of the criteria which is calculated by (40). Thus, the importance index or value of criterion is capable of being deployed as a weighting calculation method. Shapley weight provides a subjective and objective assessment of criteria in the decision making process.

6. Extended VIKOR Method with Interval-Valued Intuitionistic Fuzzy Information

In this section, we extend the VIKOR method to solve multi-criteria group decision making problems in which all preference information provided by decision makers is expressed as interval-valued intuitionistic fuzzy values, and the interaction phenomena among the preference of individual decision makers and conflicting criteria are taken into account. For multi-criteria group decision making problem, let 𝐸=(𝑒₁,𝑒₂,…,𝑒_𝑟) be the set of 𝑟 decision makers, 𝐴=(𝑎₁,𝑎₂,…,𝑎_𝑚) the set of 𝑚 alternatives, and 𝐶=(𝑐₁,𝑐₂,…,𝑐_𝑛) the set of 𝑛 criteria.

To adequately model the multi-criteria group decision making problem as above, first, the proposed interval-valued intuitionistic fuzzy Choquet integral operator is used for effectively aggregating all the opinions of individual decision makers into a collective opinion under an interval-valued intuitionistic fuzzy group decision making environment. Further, based on the VIKOR method, an interval-valued intuitionistic fuzzy VIKOR method has been developed to provide a rational, systematic decision making process by which one discovers the best solution and a compromise solution that can be used to resolve a fuzzy multi-criteria decision making problem. The extended VIKOR decision procedure of multi-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral is summarized as follows.

Step 1. For every alternative 𝑎_𝑖(𝑖=1,2,…,𝑚), each expert 𝑒_𝑘(𝑘=1,2,…,𝑟) is invited to express their individual evaluations or preferences with respect to each criterion 𝑐_𝑗(𝑗=1,2,…,𝑛) by an interval-valued intuitionistic fuzzy value 𝑎^𝑘_𝑖𝑗=([𝑎𝑎^𝑘_𝑖𝑗,𝑏𝑎^𝑘_𝑖𝑗],[𝑐𝑎^𝑘_𝑖𝑗,𝑑𝑎^𝑘_𝑖𝑗])  (𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑟), where [𝑎𝑎^𝑘_𝑖𝑗,𝑏𝑎^𝑘_𝑖𝑗] indicates the uncertain degree where expert 𝑒_𝑘 considers what the alternative 𝑎_𝑖 should satisfy the criteria 𝑐_𝑗 and [𝑐𝑎^𝑘_𝑖𝑗,𝑑𝑎^𝑘_𝑖𝑗] indicates the uncertain degree where expert 𝑒_𝑘 considers what the alternative 𝑎_𝑖 should not satisfy the criteria 𝑐_𝑗. Then we can obtain the decision making matrix as follows: 𝐷^𝑘=𝑎^𝑘₁₁,𝑎^𝑘₁₂,…,𝑎^𝑘_1𝑛𝑎^𝑘₂₁,𝑎^𝑘₂₂,…,𝑎^𝑘_2𝑛⋮𝑎^𝑘_𝑚1,𝑎^𝑘_𝑚2,…,𝑎^𝑘_𝑚𝑛. (43)

Step 2. Since the interaction phenomena among the preference of individual decision makers are considered, the 𝜆-fuzzy measure 𝑔 is used for determining the importance of each expert 𝑒_𝑘(𝑘=1,2,…,𝑟), that is, the fuzzy density 𝑔_𝑘=𝑔(𝑒_𝑘) of each expert 𝑒_𝑘. According to (4), the parameter 𝜆 of expert 𝑒_𝑘 can be calculated. Then the importance of each combination of experts can be obtained by (3).

Step 3. By (11) or Definition 7, 𝑎^𝑘_𝑖𝑗 is reordered such that 𝑎^(𝑘)_𝑖𝑗≤𝑎^(𝑘+1)_𝑖𝑗. Utilize the interval-valued intuitionistic fuzzy Choquet integral operator 𝑎_𝑖𝑗=I-IFC_𝑔𝑎¹_𝑖𝑗,…,𝑎^𝑟_𝑖𝑗=1−^𝑟_𝑘=11−𝑎𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))},1−^𝑟_𝑘=11−𝑏𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))},_𝑟𝑘=1𝑐𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝜇(𝐴(𝑘+1))},_𝑟𝑘=1𝑑𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝜇(𝐴(𝑘+1))} (44) to aggregate all the interval-valued intuitionistic fuzzy decision matrices 𝐷^𝑘=(𝑎^𝑘_𝑖𝑗)_𝑚×𝑛  (𝑘=1,2,…,𝑟) into a collective interval-valued intuitionistic fuzzy decision matrix 𝐷=(𝑎_𝑖𝑗)_𝑚×𝑛, where 𝑎_𝑖𝑗=([𝑎𝑎_𝑖𝑗,𝑏𝑎_𝑖𝑗],[𝑐𝑎_𝑖𝑗,𝑑𝑎_𝑖𝑗])(𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛), 𝐴_(𝑘)={𝑒_(𝑘),…,𝑒_(𝑟)}, 𝐴_(𝑟+1)=∅, and 𝑔(𝐴_(𝑘)) can be calculated by (3).

Step 4. Let 𝐽₁ be a collection of benefit criteria (the larger the 𝑐_𝑗, the greater the preference) and 𝐽₂ a collection of cost criteria (the smaller the 𝑐_𝑗, the greater the preference). The interval-valued intuitionistic fuzzy positive-ideal solution (IV-IFPIS), denoted by 𝛼⁺, and the interval-valued intuitionistic fuzzy negative-ideal solution (IV-IFNIS), denoted by 𝛼⁻=(𝛼⁻₁,𝛼⁻₂,…,𝛼⁻_𝑛), are defined as follows: 𝛼⁺=𝑐_𝑗,max_𝑖𝑎𝑎_𝑖𝑗,max_𝑖𝑏𝑎_𝑖𝑗∣𝑗∈𝐽₁,min_𝑖𝑎𝑎_𝑖𝑗,min_𝑖𝑏𝑎_𝑖𝑗∣𝑗∈𝐽₂,min_𝑖𝑐𝑎_𝑖𝑗,min_𝑖𝑑𝑎_𝑖𝑗∣𝑗∈𝐽₁,max_𝑖𝑐𝑎_𝑖𝑗,max_𝑖𝑑𝑎_𝑖𝑗∣𝑗∈𝐽₂=𝛼∣𝑖=1,2,…,𝑚₊₁,𝛼⁺₂,…,𝛼⁺_𝑛, (45) where 𝛼⁺_𝑗=([𝑎𝛼⁺_𝑗,𝑏𝛼⁺_𝑗],[𝑐𝛼⁺_𝑗,𝑑𝛼⁺_𝑗])  (𝑗=1,2,…,𝑛). Consider 𝛼⁻=𝑐_𝑗,min_𝑖𝑎𝑎_𝑖𝑗,min_𝑖𝑏𝑎_𝑖𝑗∣𝑗∈𝐽₁,max_𝑖𝑎𝑎_𝑖𝑗,max_𝑖𝑏𝑎_𝑖𝑗∣𝑗∈𝐽₂,max_𝑖𝑐𝑎_𝑖𝑗,max_𝑖𝑑𝑎_𝑖𝑗∣𝑗∈𝐽₁,min_𝑖𝑐𝑎_𝑖𝑗,min_𝑖𝑑𝑎_𝑖𝑗∣𝑗∈𝐽₂=𝛼∣𝑖=1,2,…,𝑚₋₁,𝛼⁻₂,…,𝛼⁻_𝑛, (46) where 𝛼⁻_𝑗=([𝑎𝛼⁻_𝑗,𝑏𝛼⁻_𝑗],[𝑐𝛼⁻_𝑗,𝑑𝛼⁻_𝑗](𝑗=1,2,…,𝑛).

Step 5. For conflicting criteria, the 𝜆-fuzzy measure 𝑔 can be used for determining the subjective assessment, that is, the importance of each criterion 𝑐_𝑗(𝑗=1,2,…,𝑛), and the fuzzy density 𝑔_𝑖=𝑔(𝑐_𝑖) of each criterion. According to (4), parameter 𝜆₂ of criteria can be determined. Then the Shapley weight of each criterion is calculated by (39) or (40).

Step 6. Inspired by the idea of VIKOR, in this step, compute 𝑆_𝑖 and 𝑅_𝑖 values for 𝑖=1,2,…,𝑚, which symbolize the average and the worst group scores for the alternative 𝐴_𝑖, respectively, with the relations as follows: 𝑆_𝑖=^𝑛_𝑗=1𝑤_𝑗×𝑎𝛼⁺_𝑗−𝑎𝑎_𝑖𝑗+𝑏𝛼⁺_𝑗−𝑏𝑎_𝑖𝑗+𝑐𝛼⁺_𝑗−𝑐𝑎_𝑖𝑗+𝑑𝛼⁺_𝑗−𝑑𝑎_𝑖𝑗𝑎𝛼⁺_𝑗−𝑎𝛼⁻_𝑗+𝑏𝛼⁺_𝑗−𝑏𝛼⁻_𝑗+𝑐𝛼⁺_𝑗−𝑐𝛼⁻_𝑗+𝑑𝛼⁺_𝑗−𝑑𝛼⁻_𝑗, (47) where 𝑆_𝑖 is interpreted as “concordance” and can provide decision makers with information about the maximum group “utility” or “majority.”
Consider 𝑅_𝑖=max_𝑗𝑤_𝑗𝑎𝛼⁺_𝑗−𝑎𝑎_𝑖𝑗+𝑏𝛼⁺_𝑗−𝑏𝑎_𝑖𝑗+𝑐𝛼⁺_𝑗−𝑐𝑎_𝑖𝑗+𝑑𝛼⁺_𝑗−𝑑𝑎_𝑖𝑗𝑎𝛼⁺_𝑗−𝑎𝛼⁻_𝑗+𝑏𝛼⁺_𝑗−𝑏𝛼⁻_𝑗+𝑐𝛼⁺_𝑗−𝑐𝛼⁻_𝑗+𝑑𝛼⁺_𝑗−𝑑𝛼⁻_𝑗, (48) where 𝑅_𝑖 is interpreted as “discordance” and provides decision makers with information about the minimum individual regret of the “opponent.”

Step 7. Compute the index values 𝑄_𝑖. These index values are defined as 𝑄_𝑖=𝑅_𝑖−𝑅⁻𝑅⁺−𝑅⁻,if𝑆⁺=𝑆⁻,𝑆_𝑖−𝑆⁻𝑆⁺−𝑆⁻,if𝑅⁺=𝑅⁻,𝛾𝑆_𝑖−𝑆⁻𝑆⁺−𝑆⁻+𝑅1−𝛾^𝑖−𝑅⁻𝑅⁺−𝑅⁻,otherwise, (49) where 𝑆⁺=Max𝑆_𝑖,𝑆⁻=Min𝑆_𝑖, which denotes that 𝑆⁻ is the maximum majority rule or maximum group utility; 𝑅⁺=Max𝑅_𝑖, 𝑅⁻=Min𝑅_𝑖, which denotes that 𝑅⁻ is the minimum individual regret of the opponent. Thus, the index 𝑄_𝑖 is based on the consideration of both the group utility and individual regret of the opponent. In addition 𝛾 is introduced as a weight for the strategy of ‘‘the majority of criteria” (or ‘‘the maximum group utility”), whereas 1−𝛾 is the weight of the individual regret. The value of 𝛾 lies in the range of 0–1 and these strategies can be compromised by 𝛾=0.5.

Step 8. Rank the alternatives, sorting by the values 𝑆, 𝑅, and 𝑄 in decreasing order. The results are three ranking lists.

Step 9. Propose as a compromise solution the alternative (𝑎⁽¹⁾) which is the best ranked by the measure 𝑄 (minimum) if the following two conditions are satisfied.
(C1) Acceptable advantage: 𝑄𝑎⁽²⁾−𝑄𝑎⁽¹⁾≥𝐷𝑄, (50) where 𝑎⁽²⁾ is the alternative with second place in the ranking list by 𝑄; 𝐷𝑄=1/(𝑚−1), 𝑚 is the number of alternatives.
(C2) Acceptable stability in decision making: the alternative 𝑎⁽¹⁾ should also be the best ranked by 𝑆 or/and 𝑅. This compromise solution is stable within the decision making process, which could be the strategy of maximum group utility (when 𝛾>0.5 is needed), or ‘‘by consensus” 𝛾≈0.5, or ‘‘with veto” (𝛾<0.5). Here, 𝛾 is the weight of decision making strategy of maximum group utility.
If one of the conditions is not satisfied, then a set of compromise solutions is proposed, which consists of:
(C3) alternatives 𝑎⁽¹⁾ and 𝑎⁽²⁾ if only the C2 is not satisfied or
(C4) alternatives 𝑎⁽¹⁾,𝑎⁽²⁾,…,𝐴^(𝑘) if the C1 is not satisfied; 𝑎^(𝑘) is determined by the relation 𝑄(𝑎^(𝑘))−𝑄(𝑎⁽¹⁾)<𝐷𝑄 for maximum 𝑘 (the positions of these alternatives are ‘‘in closeness’’).
The best alternative, ranked by 𝑄, is the one with the minimum value of 𝑄. The main ranking result is the compromise ranking list of alternatives and the compromise solution with ‘‘average rate.’’ The obtained compromise solution could be accepted by the decision makers because it provides a maximum ‘‘group utility’’ (represented by min𝑆) of the ‘‘majority,’’ and a minimum of the ‘‘individual regret’’ (represented by min𝑅) of the ‘‘opponent.’’ The compromise solutions are the basis for negotiations, involving the decision maker’s preference by criteria weights.

7. A Numerical Example

Assume that there is an investment company who wants to invest a sum of money in the best option. There is a panel with five possible alternatives to invest the money: 𝑎₁ is a car company; 𝑎₂ is a food company; 𝑎₃ is a computer company; 𝑎₄ is an arms company; and 𝑎₅ is a TV company. The investment company must take a decision according to the following four criteria: 𝑐₁ is the profit ability; 𝑐₂ is the growth analysis; 𝑐₃ is the social-political impact; and 𝑐₄ is the enterprise culture. The five possible alternatives 𝑎_𝑖  (𝑖=1,2,3,4,5) are to be evaluated using the interval-valued intuitionistic fuzzy information by three decision makers 𝑒_𝑘  (𝑘=1,2,3), as listed in the following matrix [49]:𝐷¹=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[,𝐷([0.4,0.50.3,0.4])([0.4,0.60.2,0.4])([0.1,0.30.5,0.6])([0.3,0.40.3,0.5])([0.6,0.70.2,0.3])([0.6,0.70.2,0.3])([0.4,0.70.1,0.2])([0.5,0.60.1,0.3])([0.6,0.70.1,0.2])([0.5,0.60.3,0.4])([0.5,0.60.1,0.3])([0.4,0.50.2,0.4])([0.3,0.40.2,0.3])([0.6,0.70.1,0.3])([0.3,0.40.1,0.2])([0.3,0.70.1,0.2])([0.7,0.80.1,0.2])([0.3,0.50.1,0.3])([0.5,0.60.2,0.3])([0.3,0.40.5,0.6])₂=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[,𝐷([0.3,0.40.4,0.5])([0.5,0.60.1,0.3])([0.4,0.50.3,0.4])([0.4,0.60.2,0.4])([0.3,0.60.3,0.4])([0.4,0.70.1,0.2])([0.5,0.60.2,0.3])([0.6,0.70.2,0.3])([0.6,0.80.1,0.2])([0.5,0.60.1,0.2])([0.5,0.70.2,0.3])([0.1,0.30.5,0.6])([0.4,0.50.3,0.5])([0.5,0.80.1,0.2])([0.2,0.50.3,0.4])([0.4,0.70.1,0.2])([0.6,0.70.2,0.3])([0.6,0.70.1,0.2])([0.5,0.70.2,0.3])([0.6,0.70.1,0.3])₃=],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[.([0.2,0.50.3,0.4])([0.4,0.50.1,0.2])([0.3,0.60.2,0.3])([0.3,0.70.1,0.3])([0.2,0.70.2,0.3])([0.3,0.60.2,0.4])([0.4,0.70.1,0.2])([0.5,0.80.1,0.2])([0.5,0.60.3,0.4])([0.7,0.80.1,0.2])([0.5,0.60.2,0.3])([0.4,0.50.3,0.4])([0.3,0.60.2,0.4])([0.4,0.60.2,0.3])([0.1,0.40.3,0.6])([0.3,0.70.1,0.2])([0.6,0.70.1,0.3])([0.5,0.60.3,0.4])([0.5,0.60.2,0.3])([0.5,0.60.2,0.4]) (51)

To effectively solve this problem, the proposed decision procedure as above is followed for determining the most desirable alternative as follows.

Step 1. The fuzzy density of each decision maker and its 𝜆 parameter are determined first. Suppose that 𝑔(𝑒₁)=0.40, 𝑔(𝑒₂)=0.40, and 𝑔(𝑒₃)=0.40. Parameter 𝜆 of expert can be determined: 𝜆1=−0.44. According to (3), we have 𝑔(𝑒₁,𝑒₂)=𝑔(𝑒₁,𝑒₃)=𝑔(𝑒₂,𝑒₃)=0.73, 𝑔(𝑒₁,𝑒₂,𝑒₃)=1.

Step 2. By (11) or Definition 21, 𝑎^𝑘_𝑖𝑗 is reordered such that 𝑎^(𝑘)_𝑖𝑗≤𝑎^(𝑘+1)_𝑖𝑗; utilize the interval-valued intuitionistic fuzzy Choquet integral operator 𝑎_𝑖𝑗=I-IFC_𝑔𝑎¹_𝑖𝑗,𝑎²_𝑖𝑗,𝑎³_𝑖𝑗=1−³_𝑘=11−𝑎𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))},1−³_𝑘=11−𝑏𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))},_3𝑘=1𝑐𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))},_3𝑘=1𝑑𝑎^(𝑘)_𝑖𝑗^{𝑔(𝐴(𝑘))−𝑔(𝐴(𝑘+1))} (52) to aggregate all decision maker’s interval-valued intuitionistic fuzzy decision matrices 𝐷^𝑘=(𝑎^𝑘_𝑖𝑗)_5×4  (𝑘=1,2,3) into a collective interval-valued intuitionistic fuzzy decision matrix 𝐷=(𝑎_𝑖𝑗)_5×4 as 𝐷=^{],[],[],[],[],[],[],[],[}_{],[],[],[],[],[],[],[],[],[],[],[],[}^{([0.31,0.470.32,0.42])([0.44,0.570.12,0.28])([0.29,0.500.29,0.40])([0.33,0.600.17,0.38])([0.42,0.680.22,0.32])([0.47,0.680.16,0.28])([0.43,0.680.12,0.22])([0.54,0.720.13,0.26])}_{([0.58,0.720.13,0.24])([0.59,0.700.13,0.24])([0.50,0.640.16,0.30])([0.33,0.450.29,0.45])([0.33,0.510.22,0.39])([0.51,0.720.12,0.26])([0.22,0.440.19,0.34])([0.34,0.700.10,0.20])([0.64,0.740.12,0.26])([0.50,0.620.14,0.28])([0.50,0.640.20,0.30])([0.50,0.600.19,0.40])}. (53)

Step 3. Since ([1,1],[0,0]) and ([0,0],[1,1]) are the largest and smallest interval-valued intuitionistic fuzzy values, respectively, for benefit criteria 𝑐₁,𝑐₂,𝑐₃,𝑐₄, IV-IFPIS 𝛼⁺and IV-IFNIS 𝛼⁻ can be simply denoted as follows: 𝛼⁺=],[,],[,],[,],[,𝛼(([1,10,0])([1,10,0])([1,10,0])([1,10,0]))₋=],[,],[,],[,],[.(([0,01,1])([0,01,1])([0,01,1])([0,01,1])) (54)
Alternatives 𝑎_𝑖  (𝑖=1,2,…,5) are denoted by 𝑥_𝑖=(𝑎_𝑖1,𝑎_𝑖2,𝑎_𝑖3,𝑎_𝑖4) as follows: 𝑥₁=],[,],[,],[,],[𝑥(([0.31,0.470.32,0.42])([0.44,0.570.12,0.28])([0.29,0.500.29,0.40])([0.33,0.600.17,0.38]))₂=],[,],[,],[,],[,𝑥(([0.42,0.680.22,0.32])([0.47,0.680.16,0.28])([0.43,0.680.12,0.22])([0.54,0.720.13,0.26]))₃=],[,],[,],[,],[,𝑥(([0.58,0.720.13,0.24])([0.59,0.700.13,0.24])([0.50,0.640.16,0.30])([0.33,0.450.29,0.45]))₄=],[,],[,],[,],[,𝑥(([0.33,0.510.22,0.39])([0.51,0.720.12,0.26])([0.22,0.440.19,0.34])([0.34,0.700.10,0.20]))₅=],[,],[,],[,],[.(([0.64,0.740.12,0.26])([0.50,0.620.14,0.28])([0.50,0.640.20,0.30])([0.50,0.600.19,0.40])) (55)

Step 4. The fuzzy density of each criterion and its 𝜆₂ parameter are calculated. Suppose that 𝑔(𝑐₁)=0.40, 𝑔(𝑐₂)=0.25, 𝑔(𝑐₃)=0.37, and 𝑔(𝑐₄)=0.20, according to (4); the 𝜆₂ of criteria can be determined: 𝜆₂=−0.44. According to (3), we have 𝑔(𝑐₁,𝑐₂)=0.60, 𝑔(𝑐₁,𝑐₃)=0.70, 𝑔(𝑐₁,𝑐₄)=0.56, 𝑔(𝑐₂,𝑐₃)=0.68, 𝑔(𝑐₂,𝑐₄)=0.43, 𝑔(𝑐₃,𝑐₄)=0.54, 𝑔(𝑐₁,𝑐₂,𝑐₃)=0.88, 𝑔(𝑐₁,𝑐₂,𝑐₄)=0.75, 𝑔(𝑐₂,𝑐₃,𝑐₄)=0.73, 𝑔(𝑐₁,𝑐₃,𝑐₄)=0.84, and 𝑔(𝑐₁,𝑐₂,𝑐₃,𝑐₄)=1.0. By means of (39), weighting coefficients of criteria are obtained as follows: 𝑤₁=0.32,𝑤₂=0.22,𝑤₃=0.31,𝑤₄=0.15. (56)

Step 5. According to (47)–(49), the values of 𝑅, 𝑆, and 𝑄 are calculated for all alternatives as follows. 𝑆₁=0.441,𝑆₂=0.325,𝑆₃𝑆=0.321,₄=0.397,𝑆₅𝑅=0.313₁=0.157,2003𝑅₂=0.116,𝑅₃𝑅=0.102,₄=0.145,𝑅₅𝑄=0.105,₁=1,𝑄₂𝑄=0.2545−0.1607𝛾,₃=0.0938𝛾,𝑄₄𝑄=0.7818−0.1295𝛾,₅=0.0545−0.0545𝛾. (57)

Step 6. The ranking order of the alternatives by 𝑅, 𝑆 and 𝑄 is shown in a decreasing order (see Table 1).

Table 1 

Decision result obtained from the extended VIKOR method.

Step 7. From Table 1, the decision results are obtained as follows.

The best candidate (alternative) is 𝑎₃ and the ranked ordered of all candidates is 𝑎₃≻𝑎₅≻𝑎₂≻𝑎₄≻𝑎₁ if 0≤𝛾<0.378; the best candidate is 𝑎₃ and 𝑎₅, the ranked order of all candidates is 𝑎₃∼𝑎₅≻𝑎₂≻𝑎₄≻𝑎₁ if 𝛾=0.378; the best candidate is 𝑎₅; and the ranked order of all candidates is 𝑎₅≻𝑎₃≻𝑎₂≻𝑎₄≻𝑎₁ if 0.378<𝛾<1; the best candidate is 𝑎₅, and the ranked ordered of all candidates is 𝑎₅≻𝑎₃∼𝑎₂≻𝑎₄≻𝑎₁ if 𝛾=1.

However, for 0≤𝛾≤0.018, 𝑄(𝑎⁽²⁾)−𝑄(𝑎⁽¹⁾)<1/(5−1), the set of compromise solutions is 𝑎₃ and 𝑎₅; for 0.018<𝛾≤0.378, 𝑄(𝑎⁽³⁾)−𝑄(𝑎⁽¹⁾)<1/(5−1), the set of compromise solutions is 𝑎₃, 𝑎₅, and 𝑎₂; for 0.378<𝛾≤1, 𝑄(𝑎⁽³⁾)−𝑄(𝑎⁽¹⁾)<1/(5−1), the set of compromise solutions is 𝑎₅, 𝑎₃, and 𝑎₂.

Remark 24. In [49], Tan et al. have extended TOPSIS method to solve this decision making problems with interval-valued intuitionistic fuzzy information. Here we simply make a comparison between these two methods.

Using the extended TOPSIS [49], the ranking of this example in descending order is 𝑎₅≻𝑎₂≻𝑎₃≻𝑎₄≻𝑎₁.

Mainly, the best alternative in two methods is the same. The order of the alternatives obtained by the extended TOPSIS is litter different from that obtained by the extended VIKOR. Although the extended TOPSIS method introduces two reference points, the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution, but it does not consider the relative importance of the distances from these points, which does not mean that it is always the closest to the ideal solution. In the extended VIKOR, the group majority preference to the alternative is taken into account besides individual decision-maker’s preference to the alternative. So the extended VIKOR method proposes the set of compromise solutions with an advantage rate.

8. Conclusions

VIKOR method focused on ranking and selecting from a set of alternatives in the presence of conflicting criteria and is a helpful tool in multi-criteria decision making; the obtained compromise solution could be accepted by the decision makers because it provides a maximum group utility of the majority and a minimum of the individual regret of the opponent. The fact that social, economic, technological, and environmental factors need to be taken into consideration in decision-making makes the process more complex and need multiple decision makers to make decision together. Traditional single-person decision-making is no longer able to handle these complex problems properly. Based on an intuitionistic fuzzy Choquet integral operator, in this paper, we proposed an extension of VIKOR for effectively solving the multiple criteria group decision making problem in which the interaction phenomena among preference of the decision makers are taken into account and all the criteria values are expressed in interval-valued intuitionistic fuzzy values. Subjective and objective weights of criteria are considered together, where subjective weights are obtained directly from the decision makers while objective weights are determined based on Shapley value. Using the proposed operator, a decision procedure is developed for solving the multi-criteria group decision making problem in an interval-valued intuitionistic fuzzy environment. Finally we proposed a numerical example to illustrate an application of the proposed method. It shows that the proposed procedure is capable of considering the interaction phenomena between preference of experts and conflicting criteria, which makes the decision procedure more feasible and practical for solving the multi-criteria group decision-making problem.

Acknowledgments

This work was partly supported by the Funds for Creative Research Groups of China (no. 71221061), the projects of major international cooperation NSFC (no. 71210003), the National Natural Science Foundation of China (no. 71271217, 71201089), and the Program for New Century Excellent Talents in University for the Ministry of Education of China (NCET-12-0541).